POWER. 



POAVEH. 



703 



and for the true remainder coefficients write 

 / / 



+ -^-j-j + &C., 



where a is the last power used in the quotient terms, repeated, 

 not the next one to it. Suppose, for example, we are to divide 

 x> + mj? + m-jc + o;/i 3 by 2.c' + ;,i-.t- 3m 4 . Here, since x descends and 

 m ascends regularly, we throw out x and m, and the abridged dividend 

 and divisor become 



1 + 1+1 + 5 and 2 + + 1 + 0-3 



1248 1248 



underneath which we have written the multipliers. Heuce we begin 

 with 1 + 2 + 4 + 40 and 1 + 0-2 + 0-24: 



11+2+4+40 



+ 0-2+ +24 + 48 + 48 + 864 



1+2 + 2 + 30 | + 20-21 + 48 + 864 

 Hence the quotient is 



1 - + - + 1 + S - 



and the remainder is 



20 m* _ 2J w^ 48 mf 864 m' 

 16 ~x & H? 64 I s 12S je* 



One of the easiest modifications of this rule is the division of 

 '<-"-' *-... by x p or x +p, as explained in FUACTIUNS, 



.MPOSITIOX UK. 



1'OWKll (Mechanics). The present article is not intended to enter 

 deeply into the subject, but only to remove various fallacies connected 

 with the use of the word power, which frequently perplex and confuse 

 those who attempt to study mechanics without the aid of mathe- 

 matics. 



The word poiccr has obtained a technical meaning which seems to be 

 almost peculiar to popular treatises. From among the numerous 

 combinations which occur in machinery, the lever, the inclined plane, 

 the wheel and axle, the pulley, and the screw have been selected, and 

 named mec/ianical poicert. Some have treated these as different prin- 

 ciples ; some have asserted that they are reducible to the lever and the 

 inclined plane, others to the lever only ; but it is generally asserted 

 that all mechanical contrivance is reducible to one or other of these. 

 To which of them the very powerful machine (in its way) which men 

 call a cannon is to be referred we do not know, not seeing any great 

 likeness in it either to a lever, an inclined plane, a wheel and axle, a 

 pulley, or a screw. Again, tho notions of the theories of these powers 

 are as various as those of their arrangement : some say that all are 

 creators of power, some that all are powers except the simple pulley, 

 some that none are powers, some that they are losses of power. Those 

 who deny that any of the adaptations above mentioned give power, 

 look for their meaning of the word in the action of what are thence 

 called a'jcnt*,a in the muscular strength of men and animals, the force 

 of wind, the fall of water, the expansion of steam or explosive gases, 

 4c. Admitting that all these agents arc well entitled to the name of 

 i s, it is nevertheless difficult to refuse that name to a machine. 

 Watch the effect of a large crane in unloading a vessel, compare it with 

 wh.it the same men could do by their unaided (or rather, uuadapted) 

 -th in the same time, and it will be impossible to deny that the 

 machine gives power. 



The cause of all this confusion and diversity of opinion as to the 

 way of stating facts which every one knows, arises from the word 

 power being token in two different senses, that which is true of one of 

 its meanings being untrue of another. In tho first sense, power is 

 gained whenever any thing is done quicker or better at the same 

 expense, or in as effective a manner at less expense ; whenever the 

 advantage gained, or the disadvantage avoided, is worth more than 

 the trouble and cost of the means employed. Iu the second sense, 

 lwr is gained when a new adaptation is introduced, by which au 

 existing agent is made apparently stronger. To find different phrases 

 for these different things, let us say that power in the first sense means 

 that which produces beneficial effect, and that power in the seconc 

 sense means that which produces mechanical advantage : these last 

 words have been often used by writers hi the same sense as the 

 technical word power. 



Beneficial effect may be produced in various ways, without inechan 

 ical advantage, but the benefit, as far as it is of a physical character 

 will generally be found to consist in a saving of useless labour. Thus 

 in the division of labour, than which nothing produces more of beneficia 

 effect, there is not only the moral benefit, namely, the making the 

 human agent fitter for his work by giving him a more limited range o 

 occupations, but the actual saving of the labour of laying down on 

 tool and taking up another. Again, when loaded carriages are di.-. 

 ARTS AMD SCI. DIY. VOL. VI. 



missed down an inclined plane and made to draw up the empty ones, 

 here is no gain in a mechanical point of view, for the momentum which 

 is gained by the empty carriages is lost by the loaded ones ; but the 

 momentum lost by the loaded carriages is no loss, since there is no use 

 n their delivering their contents with a great velocity, while the 

 momentum abstracted is applied to a beneficial purpose. Thirdly, a 

 imple pulley gives uo mechanical advantage whatever, since the weight 

 n one side, when the pulley is at rest, must be equal to that on the 

 ther. If however we compare the effect of this machine in raising 

 weights with the carriage of them up a ladder, we see at once a bene- 

 cial effect, amounting to a saviug of the greater part of the labour ; 

 or by using a pulley, the labourer has not to carry himself up to the 

 leight required and down again. Fourthly, when the traces by which 

 lorses draw are inclined at a proper angle, a part of the drawing 

 xjwer is taken off, and applied in lifting the carriage off the road and 

 essening the friction, so that the diminished draught is better able to 

 .o the remaining work than if the whole draught were applied to the 

 vhole friction. Here is no gain of power in the mechanical sense, 

 hough the alteration is certainly a double gain (uo matter how slight 

 a one) of beneficial effect, for the carriage is more easily drawn and the 

 oad is less worn. Numberless instances might be cited in which real 

 ifiiefit is a consequence of mere adaptation, even without the pro- 

 luction of what is called poica- in treatises on mechanics. 



In treating of the second meaning of the word power, or its synonyme, 

 nechanical advantage, we must separately consider a machine just 

 >alanced, and one in which an additional force applied gives motion. 

 Suppose a lever, one arm of which o A is ten times as long as the other 

 o B, and suppose that the arms balance each other. A pull of one pound 

 at A will then support (so it seems) ten pounds at B, for the first will 

 certainly equilibrate the second, or prevent motion. Nevertheless, it 

 s not true that one pound supports ten pounds ; nor cau one pound, 

 >y any contrivance whatsoever, be made to support more than one 

 xmnd. In the case before us, A acts against one pound, but not oue 

 )und of B ; it is A and B together, eleven pounds iu all, which oppose 

 a resistance of eleven pounds offered by the support or pivot o. Tho 



_ n 



A 



weight B is equivalent to two pressures, one downwards, at o, of 

 leven pounds, one upwards at A, of one pound. The frame on which 

 ;he pivot rests neutralises tho first, the pull at A neutralises the second. 

 To say that A supports B, would be an assertion like that of a person 

 who should say that he had paid 111. with I/., and should forget that 

 lie had borrowed 10/. elsewhere ; and in the above system there is 

 neither gain nor loss of force in any manner. When the weight B i* 

 lifted from the ground, and attached to the eud of the lever, and when 

 at the same moment the hand was applied at A, there is a pressure of 

 eleven pounds applied to the earth at o. But all this pressure was 

 first taken from the earth, ten pounds of it by the removal of the 

 weight B, and the remaining pouud by diminution of the weight of the 

 person pulling ; for a person who pulls downwards at a rope iu such a 

 manner as to supply the place of a pound weight at the cud of the 

 rope, lessens his weight by one pound. 



In the preceding manner it may be shown that every machine iu 

 which A, the less, balances B, the greater, is a case of the following 

 kind : B is equivalent to certain forces, p, <j, u, &c., applied at various 

 points of the system, and to c, equal and opposite to A. Imagine 

 p, Q, K, &c. and c substituted for B, then v, Q, B, &c. are counter- 

 balanced by the resistance of the parts of the machine, and c, and c 

 only, ia counterbalanced by A. It may also be shown that the whole 

 pressure upon the earth is just what it would be if the machine were 

 dismantled, and its parts, together with the weights, laid upon the 

 ground. 



Let us now consider the machine in motion. The paradox about 

 power here is, that the smaller weight is made to lift the greater ; for 

 example, that a pressure, say of one pound, applied to the handle of a 

 crane, lifts a weight say of twenty pounds. As long as no more precise 

 mode of expression is employed, the paradox continues : the smaller 

 weight does lift the greater. To take the simplest case, suppose 

 that the descent of a smaller weight causes the ascent of a greater, 

 as in an inclined plane with a pulley. It is now obvious that 

 before the larger weight B can rise 

 through A c, the smaller weight A must 

 fall through a length equal to B c. In 

 any machine in wlu'ch th hand, exerting 

 a pressure of one pound, balances a. 



weight of twenty pounds, the hand -*= ~^~~^~ 



when put in motion to raise the weight, will move twenty times as fast 

 as the weight. It is then oue pouud moved through twenty feet, 



z /. 



